**R E S E A R C H**

**Open Access**

## On some power series with algebraic

## coefﬁcients and Liouville numbers

### Gül Karadeniz Gözeri

**_{Correspondence:}
gulkaradeniz@gmail.com
Department of Mathematics,
Faculty of Science, Istanbul
University, Vezneciler, Istanbul,
Turkiye

**Abstract**

In this work, we consider some power series with algebraic coeﬃcients from a certain
algebraic number ﬁeld*K*of degree*m*and investigate transcendence of the values of
the given series for some Liouville number arguments.

**1 Introduction**

The theory of transcendental numbers has a long history and was originated back to Li-ouville in his famous paper [] in which he produced the ﬁrst explicit examples of tran-scendental numbers at a time where their existence was not yet known. Later, Cantor [] gave another proof of the existence of transcendental numbers by establishing the denu-merability of the set of algebraic numbers. It follows from this that almost all real numbers are transcendental. Further, the theory of transcendental numbers is closely related to the study of Diophantine approximation. Recent advances in Diophantine approximation can be found in the excellent surveys of Moshchevitin [] and Waldschimidt [].

Mahler [] introduced a classiﬁcation of the set of all transcendental numbers into three
disjoint classes, termed*S*,*T* and*U* and this classiﬁcation has proved to be of
consider-able value in the general development of the subject. The ﬁrst classiﬁcation of this kind
was outlined by Maillet in [], and others were described by Perna in [] and
Morduchai-Boltovskoj [] but to Mahler’s classiﬁcation attaches by for the most interest. Mahler
de-scribed this classiﬁcation in the following way.

Let*P*(*x*) =*anxn*+· · ·+*ax*+*a*be a polynomial with integral coeﬃcients. The height
*H*(*P*) of*P*is deﬁned by*H*(*P*) =max(|*an*|, . . . ,|*a*|) and the degree of*P*is denoted bydeg(*P*).
Given an arbitrary complex number*ξ*, Mahler puts

*ωn(H*,*ξ*) =min*P*(*ξ*):deg(*P*)≤*n*,*H*(*P*)≤*H*,*P*(*ξ*)= ,
where*n*and*H*are positive integers. Next Mahler puts

*ωn*(*ξ*) =* _{H→∞}*lim

–log*ωn(H*,*ξ*)

log*H* and *ω*(*ξ*) =*n→∞*lim
*ωn(ξ*)

*n* .

The inequalities ≤*ωn(ξ*)≤ ∞and ≤*ω*(*ξ*)≤ ∞hold. From*ωn*+(*H*,*ξ*)≤*ωn(H*,*ξ*), we
get*ωn*+(*ξ*)≥*ωn(ξ*). If for an index*ωn(ξ*) =∞, the*μ*(*ξ*) is deﬁned as the smallest of them,
otherwise*μ*(*ξ*) =∞. Thus,*μ*(*ξ*) is uniquely determined. Furthermore, the two quantities
*μ*(*ξ*) and*ω*(*ξ*) are never ﬁnite simultaneously, for the ﬁniteness of*μ*(*ξ*) implies that there

is an *n*<∞such that*ωn*=∞, whence*ω*=∞. Therefore, there are the following four
possibilities for*ξ*,*ξ*is called

an*A*- number if*ω*(*ξ*) = ,*μ*(*ξ*) =∞,
an*S*- number if <*ω*(*ξ*) <∞,*μ*(*ξ*) =∞,
a*T*- number if*ω*(*ξ*) =∞,*μ*(*ξ*) =∞,
a*U*- number if*ω*(*ξ*) =∞,*μ*(*ξ*) <∞.

In [], Koksma introduced an analogous classiﬁcation of complex numbers. He divided
the complex numbers into four classes*A*∗,*S*∗,*T*∗and*U*∗in the following way.

Let*α*be an arbitrary algebraic number. If we denote its minimal deﬁning polynomial by
*P*(*x*), then the height*H*(*α*) of*α* is deﬁned by*H*(*α*) =*H*(*P*) and the degreedeg(*α*) of*α*is
deﬁned bydeg(*α*) =deg(*P*). Given an arbitrary complex number*ξ* and positive integers*n*,
*H*, let*α*be an algebraic number with degree at most*n*and height at most*H*such that|*ξ*–*α*|
takes the smallest positive value; Koksma deﬁnes*ω*∗* _{n}*(

*H*,

*ξ*) by the following equation:

*ω*∗* _{n}*(

*H*,

*ξ*) =min|

*ξ*–

*α*|:

*α*is algebraic,deg(

*α*)≤

*n*,

*H*(

*α*)≤

*H*,

*α*=

*ξ*. Next, Koksma puts

*ω*∗* _{n}*(

*ξ*) = lim

*H→∞*

–log(*Hω*∗* _{n}*(

*H*,

*ξ*))

log*H* and *ω*

∗_{(}_{ξ}_{) =} _{lim}
*n→∞*

*ω*∗* _{n}*(

*ξ*)

*n* .

The inequalities ≤*ω*∗* _{n}*(

*ξ*)≤ ∞and ≤

*ω*∗(

*ξ*)≤ ∞hold. If for an index

*ω*∗

*(*

_{n}*ξ*) =∞, the

*μ*∗(

*ξ*) is deﬁned as the smallest of them, otherwise

*μ*∗(

*ξ*) =∞. Thus,

*μ*∗(

*ξ*) is uniquely determined. Furthermore, the two quantities

*μ*∗(

*ξ*) and

*ω*∗(

*ξ*) are never ﬁnite simultane-ously. Therefore, there are the following four possibilities for

*ξ*,

*ξ*is called:

an*A*∗- number if*ω*∗(*ξ*) = ,*μ*∗(*ξ*) =∞,
an*S*∗- number if <*ω*∗(*ξ*) <∞,*μ*∗(*ξ*) =∞,
a*T*∗- number if*ω*∗(*ξ*) =∞,*μ*∗(*ξ*) =∞,
a*U*∗- number if*ω*∗(*ξ*) =∞,*μ*∗(*ξ*) <∞.

sums. In [], Nyblom employed a variation on the proof used to established Liouville’s theorem concerning the rational approximation of algebraic numbers, to deduce explicit growth conditions for a certain series to converge to a transcendental number. Later, Ny-blom [] derived a suﬃciency condition for a series of positive rational terms to converge to a transcendental number. Further, Duverney [] proved a theorem that gives a crite-rion for the sums of inﬁnite series to be transcendental. The terms of these series consist of the rational numbers and converge regularly and very quickly to zero. In [], Hančl introduced the concept of transcendental sequences and proved a criterion for sequences to be transcendental. Later, a new concept of a Liouville sequence was introduced in [] by means of the related Liouville series. Some recent results for the transcendence of inﬁ-nite series can also be found in Borwein and Coons [], Hančl and Rucki [], Hančl and Štěpnička [], Murty and Weatherby [], Weatherby [].

In the present work, we considered certain power series with algebraic coeﬃcients from
a certain algebraic number ﬁeld*K*of degree*m*and showed that under certain conditions
these series take values belonging to either the algebraic number ﬁeld*K* or*m _{i}*

_{=}

*Ui*in Mahler’s classiﬁcation of the complex numbers for some Liouville number arguments.

**2 Preliminaries**

In this paper, |*x*| means the absolute value of *x* and the least common multiple of
*x*,*x*, . . . ,*xn*is denoted by [*x*,*x*, . . . ,*xn].*

**Deﬁnition ** A real number*ξ*is called a Liouville number if and only if for every positive
integer*n*there exists integers*pn*,*qn*(*qn*> ) with

<*ξ*–*pn*
*qn*

<

*qn*
*n*
.

The set of all Liouville numbers is identical with the *U* subclass. More information
about Liouville numbers may be found in [–]. Now, in order to prove our main
the-orem we need the following lemmas.

**Lemma **[] *Letα*, . . . ,*αk* (*k*≥)*be algebraic numbers which belong to an algebraic*
*number ﬁeld K of degree m*, *and let F*(*y*,*x*, . . . ,*xk)be a polynomial with rational *
*inte-gral coeﬃcients and with degree at least* *in y*. *Ifηis any algebraic number such that*
*F*(*η*,*α*, . . . ,*αk*) = ,*then*deg(*η*)≤*dm and*

*H*(*η*)≤*dm*+(*l*+···+*lk*)*mHmH*(*α*)*l**m*· · ·*H*(*αk)lkm*,

*where H is the height of the polynomial F*,*d is the degree of F in y and liis the degree of F*
*in xifor i*= , . . . ,*k*.

**Lemma **[] *Letαbe an algebraic number of degree m*,*and letα*()=*α*, . . . ,*α*(*m*)*be its*
*conjugates*.*Then*|*α*| ≤*H*(*α*),*where*|*α*|=max(|*α*()|, . . . ,|*α*(*m*)|).

**Lemma **[] *Letα* *be an algebraic number of degree m*,*then H*(*α*)≤(|*α*|)*m*,*where*

**3 Main result**

**Theorem ** *Let K be an algebraic number ﬁeld of degree m*,*and let*

*g*(*x*) =
∞

*n*=

*αn*
*en*

*xn*

*be a power series such thatαn*∈*K are non-zero algebraic numbers and en*> *are rational*
*integers satisfying the following conditions*:

lim

*n→∞*

log*en*+

log*en*

=*η*> , ()

lim

*n→∞*

log*en*+

log*en*

=∞, ()

lim

*n→∞*

log*H*(*αn*)

log*en*

=*μ*< . ()

*Further*,*letξbe a Liouville number satisfying the following two properties*:

. *ξhas an approximation with rational numbers* *pn _{qn}* (

*qn*> )

*so that the following*

*inequality holds for suﬃciently largen*

*ξ*–*pn*

*qn*

<

*qnsnn*

lim

*n→∞sn*= +∞

. ()

. *There exist two positive real constantsγ**andγ**with* _{η}η_{–}<*γ*<*γ**and*

*eγ*

*n* ≤*qnn*≤*e*

*γ*

*n* ()

*for suﬃciently largen*.

*Then g*(*ξ*)*belongs to either the algebraic number ﬁeld K orm _{i}*

_{=}

*Ui*.

*Proof* It follows from () that

log*en*+>*η*log*en* ()

for suﬃciently large*n*, where*η*=*η*–*ε*and*ε*is to be chosen as <*ε*<*η*–*γγ*– . It follows
from () that the sequence{*en*}is strictly increasing, thus we have

lim

*n→∞en*=∞, ()

lim

*n→∞*

log*en*

*n* =∞ and *n→∞*lim

log*en*

*n* =∞. ()

Furthermore, from () we get

*en*<*e*

*η*_{}

*n*+ ()

Let*En*= [*e*,*e*, . . . ,*en]. Then by using () and (), we obtain for suﬃciently largen*

*en*≤*En*≤*e*

*ε*+ *η*

*η*–

*n* , ()

where*ε*is to be chosen as <*ε*<*γ*–_{η}_{–}*η* . We can easily deduce from () and*μ*< *μ*_{}+
that

log*H*(*αn*)

log*en*

<*μ*+

()

for suﬃciently large*n*. Since () holds, there is a natural number*N*such that

*H*(*αn) <e*(

*μ*+
)

*n* ()

for every*n*>*N*. On the other hand, we get from Lemma that

|*αn*| ≤*H*(*αn)* ()

since*αn*are algebraic numbers. From here and (), we obtain

|*αn*| ≤ |*αn*| ≤*e*
(*μ*+_{} )

*n* ()

for every*n*>*N*. Now, we shall deﬁne the algebraic numbers

*βn*=
*n*

*γ*=

*αγ*

*eγ*

*pn*
*qn*

γ

for*n*= , , , . . . . Since*βn*∈*K*,deg(*βn*)≤*m*for*n*= , , , . . . . Let us determine an
up-per bound for the heights of the algebraic numbers*βn*. By multiplying both sides of this
equality by*Enqnn*and putting*li*=*En _{ei}* for

*i*= , , , . . . , we obtain the equality

*Enqnnβn*=

*lqn _{n}α*+

*lqn _{n}*–

*pn*

*α*+· · ·+

*lnpnn*

*αn.*

Since*ξ* is a Liouville number, we can assume that*pn*= for*n*= , , , . . . . Then we get a
polynomial

*G*(*y*,*x*,*x*, . . . ,*xn) =Enqnny*–
*n*

*γ*=

*lγqn _{n}*–

*γpγ*

_{n}*xγ*

with rational integral coeﬃcients such that*G*(*βn,α*, . . . ,*αn) = . Further, this polynomial*
is of degree in each*y*,*x*,*x*, . . . ,*xn*. Thus, we deduce from Lemma that

*H*(*βn)*≤*m*+(*n*+)*mHmH*(*α*)*m*· · ·*H*(*αn)m*, ()
where*H*is the height of the polynomial*G*(*y*,*x*,*x*, . . . ,*xn*). By using (), we obtain|*pn _{qn}*|

*i*≤

*kn*

for*i*= , , , . . . , where*k*=|*ξ*|+ > is a real constant. From here, we can easily get

It follows from () and () that

*H*(*βn)*≤*k*_{}*mnEm _{n}qmn_{n}*

*H*(

*α*)

*m*· · ·

*H*(

*αn)m*, ()

where*k*= *k*> is a real constant. Moreover, we get

*H*(*βi*)≤

|*βi*|

*m*

(*i*= , , , . . .) ()

from Lemma . Then we deduce from () and () that

*H*(*βn)*≤*m*(*n*+)*k*_{}*mnEm _{n}qmn_{n}* |

*β*| · · · |

*βn*|

*m*

. ()

By using (), () and (), we obtain

|*β*| · · · |*βn*|

*m*

≤*m*(*n*+)*keγ**m*
_{(}*μ*+

)

*n* ,

where*k*=max(, (*H*(*α*)· · ·*H*(*αN*))

*m*_{)}_{≥}_{ is a real constant. It follows from here, () and}
() that

*H*(*βn)*≤*k*_{}*nek*

*n* , ()

where*k*= *m**k*_{}*mk*> and*k*=*γ*m+*γ*m(*μ*_{}+) +*γ*m> are real constants.
Now, we consider the following polynomials:

*gn*(*x*) =
*n*

*ν*=

*αν*

*eν*

*xν*

for*n*= , , . . . . Since*gn*(*x*) are continuous and diﬀerentiable for all real numbers, at least
one real number*cn*exists between*ξ* and*pn _{qn}* such that for every

*n*

*gn*(*ξ*) –*βn*=*gn*(*ξ*) –*gn*
*pn*
*qn*

= *ξ*–*pn*
*qn*

*g _{n}*(

*cn*). ()

It is obvious that|*cn*| ≤max(|*ξ*|,|*pn _{qn}*|). Since|

*pn*|<|

_{qn}*ξ*|+ , we obtain|

*cn*| ≤ |

*ξ*|+ for suﬃ-ciently large

*n*. Furthermore, from here and () and (), we get

*gn*(*ξ*) –*gn*
*pn*
*qn*

≤

*qnnsn*
*n*

*ν*=

|*αν*|

*eν*

*ν*|*ξ*|+ ν– ()

for suﬃciently large*n*.

Deﬁne*σn*=max(|*α*|, . . . ,|*αn*|). Then we obtain
*n*

*ν*=

|*αν*|

*eν*

*ν*|*ξ*|+ ν–≤*n**σn*

for suﬃciently large*n*. It follows from () that*σn*≤*e*
(+_{}*μ*)

*n* for suﬃciently large*n*. We get
from here and (), ()

*gn(ξ*) –*βn*≤

*n*(|*ξ*|+ )*n*–* _{e}*(+

*μ*)

*n*

*qnsnn*

.

By using (), we obtain from here that

*gn(ξ*) –*βn*≤

*n*(|*ξ*|+ )*n*–
*eγ**sn*–(

+*μ*

)
*n*

. ()

From () andlim* _{n→∞}sn*=∞, it is possible to ﬁnd a sequence{

*ωn*}withlim

*n→∞ωn*=∞ such that

*n*_{(}_{|}_{ξ}_{|}_{+ )}*n*–
*eγ**sn*–(

+*μ*

)
*n*

≤

(*kn*_{}*ek*
*n*)*ωn*

. ()

Therefore, we get from (), () and ()

*gn*(*ξ*) –*βn*≤

*H*(*βn)ωn* ()

for suﬃciently large*n*. Moreover, the following inequality holds:

*g*(*ξ*) –*gn(ξ*)≤
∞

*i*=

|*αn*+*i*|
*en*+*i* |

*ξ*|*n*+*i*.

We get from here and ()

*g*(*ξ*) –*gn(ξ*)≤|*ξ*|
*n*+

*e*(
–*μ*

)
*n*+

+ *en*+
*en*+

–*μ*

|*ξ*|+ *en*+
*en*+

–*μ*

|*ξ*|+· · ·

.

Thus, we deduce from () that

< *en*+
*en*+

<
*e*(–

*η*)
*n*+

.

Since () holds, then we obtainlim*n→∞en _{en}*+

_{+}= . Similarly, since –

_{}

*μ*> , we have

*en*+
*en*++*k*

–*μ*

|*ξ*|*k*<

*k*

for*k*= , , , . . . and, therefore,

*g*(*ξ*) –*gn(ξ*)≤|*ξ*|
*n*+

*e*(
–*μ*

)
*n*+

On the other hand from (), we get |*ξ*|*n*+_{≤}* _{e}*(–

*μ*)

*n*+ for suﬃciently large*n*. From here, we
obtain

*g*(*ξ*) –*gn*(*ξ*)≤

*e*(
–*μ*

)
*n*+

for suﬃciently large*n*. Now if we deﬁne*rn*=loglog*enen*+, then we have

*g*(*ξ*) –*gn*(*ξ*)≤

*ern*(
–*μ*

)
*n*

.

Using (), then it follows that there exists a subsequence{*rnk*}of{*rn*}such thatlim*k→∞rnk*=

∞. Therefore, we get

*g*(*ξ*) –*gnk*(*ξ*)≤

*ernk*(
–*μ*

)
*nk*

()

for suﬃciently large*nk. From () and*lim*k→∞rnk*=∞, there exists a suitable sequence{*rnk*}
withlim* _{k→∞}r_{nk}*=∞such that

*ernk*(
–*μ*

)
*nk*

≤

(*kn*
*e*

*k*
*n*)*rnk*

and, therefore, from () and (), we obtain

*g*(*ξ*) –*gnk*(*ξ*)≤

*H*(*βnk*)*rnk*

()

for suﬃciently large*nk. On the other hand by using (), we deduce that*

* _{g}_{nk}*(

*ξ*) –

*βnk*≤

*H*(*βnk*)*ωnk*

()

for suﬃciently large*nk*. Let*ωnk*=min(*rnk*,*ωnk*). It follows from () and () that

*g*(*ξ*) –*βnk*≤
*H*(*βnk*)*ωnk*

,

wherelim*k→∞ωnk*=∞. It follows from here thatlim*k→∞βnk*=*g*(*ξ*). Thus, if the sequence

{*βnk*}is constant, then*g*(*ξ*) is an algebraic number in*K*. Otherwise*g*(*ξ*)∈

*m*

*i*=*Ui.*

**4 Conclusion**

**Competing interests**

The author declares that she has no competing interests.

**Acknowledgements**

Dedicated to Professor Hari M Srivastava.

The author would like to express her sincere thanks to the referees for their careful reading and for making some valuable comments, which have essentially improved the presentation of this paper.

Received: 14 December 2012 Accepted: 2 April 2013 Published: 16 April 2013

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doi:10.1186/1029-242X-2013-178

**Cite this article as:**Karadeniz Gözeri:On some power series with algebraic coefﬁcients and Liouville numbers.